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For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist.

ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏⵏⵉⴹⵏ, ⵣⵓⵏⴷ ⵡⵉⵙⵙ 5 ⵎⵙⴰⵙⴰⵏ ⵉⵎⴰⵙⵙⴰⵏ ⵙ ⵜⵣⴰⵢⴽⵓⵜ ⵖⴼ ⵢⵓⵡⵏ ⵓⵙⵙⴼⵔⵓ, ⵉⵜⵜⵓⴼⴽ ⵓⴼⵙⵙⴰⵢ ⵉ ⵓⵙⵙⴼⵔⵓⵜⵏ ⵉⵜⵢⴰⵙⵜⴰⵢⵏ, ⵎⴰⴽⴰ ⵍⵍⴰⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵉⵍⴰ ⵜⴰⵣⵍⴰⵖ ⵣⴰⵕⵙ ⵓⵔ ⵜⴰ ⵜⵜⵢⴰⴼⵙⴰⵢⵏ.

There are two problems that are not only unresolved but may in fact be unresolvable by modern standards.

ⵍⵍⴰⵏ ⵙⵏⴰⵜ ⵜⵎⵓⴽⵔⵉⵙⵉⵏ ⴷⴰⵢ ⵓⵔⵜⴰ ⵜⵢⴰⴼⵙⴰⵢⵏⵜ, ⵎⴰⴽⴰ ⴳ ⵜⵉⵏⴰⵡⵜ ⵉⵖⵢ ⵉⵙ ⵓⵔ ⵍⵉⵏⵜ ⵉⴼⵙⵙⴰⵢⵏ ⵙ ⵉⵏⴰⵡⴰⵢⵏ ⴰⴷ ⵉⵜⵔⴰⵔⵏ.

The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance.

ⵜⵜⵓⴼⴽ ⴰⵙⵏ ⵜⴰⵖⴷⴼⵜ ⴽⵉⴳⴰⵏ ⵉ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴰⵏ ⴷ ⵉⵇⵉⵎⴰⵏ ⵙⴳ ⵓⴳⵏⴰⵔ ⴷ ⵙⵎⵎⵓⵙ, ⴳ ⵜⵢⵉⵔⵉⵡⵉⵏ ⵏ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ ⵙⵉⵎⵔⴰⵡ ⵜⵙⵓⵍ ⵜⵡⵓⵔⵉ ⵖⴼ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴰⴷ ⵜⵍⴰ ⵜⴰⵖⴹⴼⵜ.

Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.

ⵉⴷⴷⵔ ⵀⴰⵍⴱⵔⵜ 12 ⵏ ⵓⵙⴳⴳⵯⴰⵙ, ⴹⴰⵕⵜ ⵓⴼⵙⴰⵔ ⵏ ⴽⵓⵔⵜ ⴳⵓⴷⵍ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ, ⵎⴰⴽⴰ ⵓⵔ ⵉⴱⴰⵢⵏ ⵉⵙ ⵢⴰⵔⵓ ⴽⴰⵏ ⵜⵎⵔⴰⵔⵓⵜ ⵖⴼ ⵜⵡⵓⵔⵉ ⴳⵓⴷⵍ.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.

ⵉⴳ ⵉⵍⵍⴰ ⵓⵎⵔⴰⵔⴰ ⵏ ⵡⴰⵡⴰⵍ ⵖⴼ ⵜⴰⵏⵏⴰⵢⵜ ⵏⵏⵙ ⵏ ⴽⵓ ⵜⴰⵎⵓⴽⵔⵉⵙⵜ ⵜⵓⵙⵏⴰⴽⵜ ⵉⵇⵏⴻⵏ ⴰⵙ ⵢⵉⵍⵉ ⵓⴼⵙⵙⴰⵢ, ⵢⵓⴷⵊⴰ ⵀⵉⵍⴱⵔ ⵜⴰⵣⵎⵔⵜ ⵏ ⵡⵉⵙ ⵉⵖⵢ ⵓⴼⵙⵙⴰⵢ ⴰⴷ ⵉⴳ ⵉⵏⵉⴳⵉ ⵏ ⵡⴰⴳⵓⵎ ⵏ ⵜⵎⵓⴽⵔⵉⵙⵜ ⵜⴰⵥⵓⵕⴰⵏⵜ.

The first of these was proved by Bernard Dwork; a completely different proof of the first two, via ℓ-adic cohomology, was given by Alexander Grothendieck.

ⵜⵜⵓⵡⵔ ⵜⵎⵣⵡⴰⵔⵓⵜ ⵏⵏⵙⵏ ⵙⴳ ⵖⵓⵔ ⴱⵉⵔⵏⴰⵔⴷ ⴷⵓⵔⴽ, ⵉⴼⴽ ⴰⵍⵉⴽⵙⴰⵏⴷⵔ ⴳⵔⵓⵜⵉⵏⴷⵉⴽ ⴰⵏⵥⴰ ⵓⵔ ⵢⴰⴽⵙⵓⵍⵏ ⴰⴽⴽⵯ ⴷ ⵉⵎⵣⵡⵓⵔⴰ, ⴳ ⵜⵓⵙⵙⵏⴰⵎⵢⵉⴷⵉⵔⵜ ⵉⵛⵛⴰⵔⵏ ℓ-adic.

However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴳⴰⵏ ⵉⵙⵡⵉⵏⴳⵉⵎⵏ ⵏ ⵡⵉⵍ; ⴳ ⵓⵙⴰⵜⴰⵍ ⵏⵏⵙⵏⵜ ⵣⵓⵏⴷ ⵜⴰⵎⵓⴽⵔⵉⵙⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵏ ⵀⵉⵍⴱⵔ, ⴷ ⵓⵔ ⵉⵏⵏⵉ ⵡⵉⵍ ⵉⵙ ⵉⴷ ⵏⵜⵜⴰⵜ ⴰⵢⴷ ⵉⵏⵏⴰⵏ ⴰⴷ ⵉⴳ ⴰⵖⴰⵡⴰⵙ ⵏ ⵜⵓⵙⵏⴰⴽⵉⵏ ⴽⵓⵍⵍⵓ.

Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.

ⴰⴽⴽⴰⵜ ⴰⵔ ⵡⴰⵍⴰ ⵢⴰⴽⴽⴰ ⵉⵔⴷⵓⵙ ⵜⵉⵙⵎⵖⵓⵔⵉⵏ ⵏ ⵉⴷⵔⵉⵎⵏ, ⴰⵜⵉⴳ ⵏ ⵜⵙⵎⵖⵓⵔⵜ ⵖⴼ ⵛⵇⵇⵉⵢⵜ ⵏ ⵜⵎⵓⴽⵔⵉⵙⵜ ⵉⵜⵢⴰⵏⵏⴰⵢⵏ.

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute.

ⵡⴰⵅⵅⴰ ⴰⵢⵏⵏⴰⵖ ⴳ ⵉⵎⴰⵙⵙⵏ ⵏ ⵓⵙⵎⵎⴰⵍ ⵉⴷⵙⵍⴰⵏ, ⵜⴳⴰ ⵜⴰⵔⵡⵙⴰⵏⵜ ⵜⵉⵎⴳⴳⵉⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⵍⴱⵔⵜ ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 21, ⵜⴰⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵜⵙⵎⵖⵓⵔⵜ ⵏ ⵜⵉⴼⴹⵜ ⵜⵉⵙⵙ ⵙⴰ ⵉⵙⵜⵉ ⵓⵙⵉⵏⴰⴳ ⴽⵍⴰⵢ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 2000.

The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.

ⵜⵓⵔⴷⴰ ⵏ ⵔⵉⵎⴰⵏ ⵜⵔⴰ ⴰⵙⵏⵉⵖⵙ ⴰⵛⴽⵓ ⵜⴱⴰⵢⵏⴷ ⴳ ⵜⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⴱⵔⵜ, ⴷ ⵜⵍⴳⴰⵎⵜ ⵏ ⵙⵎⵉⵍ, ⴷ ⵜⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵓⵙⵎⵖⵔ ⵏ ⵜⵉⴼⴹⵏⵜ, ⵓⵍⴰ ⴰⵡⴷ ⴳ ⵉⵙⵡⵉⵏⴳⵉⵎⵏ ⵏ ⵡⵉⵍ ⵙⴳ ⵜⴰⵍⵖⴰ ⵏⵏⵙ ⵜⴰⵏⵣⴳⴰⵏⵜ.

1931, 1936 3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?

1931, 3 ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1936 ⵉⴳ ⵏⵓⵙⵢ ⵙⵉⵏ ⵉⴼⵔⵖⴰⵙ ⵢⴰⴽⵙⵓⵍⵏ ⴳ ⵓⴽⵙⴰⵢ, ⵉⵙ ⵏⵥⴹⴰⵕ ⴰⴷ ⴰⵀⴰ ⵏⴱⵟⵟⵓ ⴰⵎⵣⵡⴰⵔⵓ ⴰⵔ ⴷ ⵉⴳ ⵜⴳⵣⵣⵓⵎⵉⵏ ⵜⵉⵎⵥⵍⴰⵢ ⵉⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⴷⵍⴰⵙⵏ, ⵏⵏⴰ ⵙ ⵏⵖⵢ ⴰⴷⵜⵏ ⵏⵙⵎⵓⵏ ⴰⵔ ⴷ ⴰⵖ ⴷ ⴽⵉⵏ ⵜⵉⵙⵙ ⵙⵏⴰⵜ?

— 12th Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field.

__12 ⵉⵙⵙⴰⵔⵡ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⵡⵉⴱⵔ ⵜⵢⵓⵔⵎ ⵅⴼ ⵉⵣⴷⴰⵎⵏ ⵏ ⴰⴱⵉⵍⵢⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ ⵖⵔ ⴽⴰ ⵉⴳⴰⵜ ⵉⴳⵔ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⴷⵙⵍⴰⵏ.

1959 15th Rigorous foundation of Schubert's enumerative calculus.

15 ⴳ 1959 ⵜⴰⵙⵉⵍⴰ ⵜⵓⵇⵊⵉⵕⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⵛⵓⴱⵔⵜ ⴰⵎⵉⴹⴰⵏ.

1927 18th (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?(b) What is the densest sphere packing?

18 ⴳ 1927 (ⴰ) ⵉⵙ ⵉⵍⵍⴰ ⴱⵓ ⵉⵎⵢⴰⵏⴰⵡⵏ ⵉⴷⵍⴰⵙⵏ ⵉⵜⵜⴰⴷⵊⴰⵏ ⴰⴽⴼⴰⴼ ⵎⵉ ⵜⵢⴰⴽⵣⵏ ⵉⴷⵍⴰⵙⵏ ⵏⵏⵙ ⴳ ⴽⵕⴰⴹ ⵡⵓⴳⴳⵓⴳⵏ? (ⴱ) ⵎⴰⵜⵜⴰ ⵉⴳⵔⴰⵏ ⴳ ⵜⴳⴳⵓⴷⵢ ⵜⴰⵏⵥⵥⵉ ⴳ ⵓⵙⵎⵓⵙⵙⵓ?

A number is a mathematical object used to count, measure, and label.

ⵓⵟⵟⵓⵏ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵖⵏⴰⵡ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⴳ ⵓⵙⵙⵉⵟⵏ ⴷ ⵓⵙⵖⴰⵍ ⴷ ⵓⵙⴻⵙⵙⴰⵖ.

"More universally, individual numbers can be represented by symbols, called numerals; for example, ""5"" is a numeral that represents the number five."

ⵙ ⵓⵎⴰⵜⴰ, ⵉⵖⵢ ⵓⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵥⵍⴰⵢ ⵙ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵉⴳⴰⵏ ⵉⵏⵎⴹⴰⵏ, ⵙ ⵓⵎⴷⵢⴰ “5” ⵜⴳⴰ ⵉⵎⵉⴹ ⴰⵖ ⵢⴰⴽⴽⴰⵏ ⵓⵟⵟⵓⵏ ⵙⵎⵎⵓⵙ.

Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation.

ⴰⵙⵙⵉⵟⵏ ⵙ ⵡⵓⵟⵟⵓⵏ ⴷⴰ ⵜⵡⴰⴼⴽⴰ ⵙⴳ ⵜⵎⵀⵍⵉⵏ ⵏ ⵓⵙⵙⵉⵟⵏ, ⵜⵉⵏⵏⴰ ⵡⴰⵍⴰ ⵉⵍⵍⴰⵏ; ⴰⵎⴰⴳⵓⵜ ⴷ ⵜⵓⴽⴽⵙⴰ, ⴷ ⵓⵙⴼⵓⴽⵜⵉ ⴷ ⵜⵓⵟⵟⵓⵜ ⴷ ⵜⴰⵙⵉⵍⴰ.

Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24, 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, Nov 30, 1992.

ⴳⵉⵍⵙⴷⵓⵔⴼ, ⵜⵓⵎⴰⵙ ⵉⵢ, ⵜⴰⵎⵙⵙⵏⴽⴷⵜ ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⵜⴰⴷⵍⵙⴰⵏⵜ, ⴳ ⵜⵣⵔⴰⵡⵜ ⵏ ⵓⵟⵓⵎⵢⵉ ⴷ ⵉⵏⴽⴰⵙ, ⵊⵓⵏ ⵡⵉⵍⵉⵢ & ⵙⵓⵏⵙ, 24 ⵉⴱⵔⵉⵔ 2012. ⵔⵉⵙⵜⵉⴼⵓ , ⵙ. ⵜⵓⵙⵏⴰⴽⵜ ⴳ ⵓⵖⵔⴼ ⴷ ⵓⵎⵣⵔⵓⵢ, ⵙⴱⵔⴰⵏⴳⵔ ⵙⴰⵢⵙ, ⵉⵎⴰⵙⵙⵏ ⵏ ⵓⵙⵏⵖⵎⵙ ⴰⵙⴱⴱⴰⴱ, 30 ⵏⵓⵡⴰⵏⴱⵉⵔ 1992.

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept.

ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 19, ⵙⵙⵏⵜⵉⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴰⵙⴱⵓⵖⵍⵓ ⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵡⵏⴳⵉⵎⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ ⵓⵛⵓⵔⵏⵜ ⵉⵜⵙⵏ ⵉⵏⵥⵍⴰⵢⵏ ⵏ ⵡⵓⵟⵟⵓⵏ, ⵏⵖ ⴰⴷ ⵏⵉⵏⵉ ⴰⵙⵙⵉⵔⵡ ⵏ ⵓⵙⵉⵙⵙⵏ.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers.

ⴰⵏⴳⵔⴰⵡ ⵏ ⵓⵙⵎⵣⴰⵣⴰⵍ ⵓⵔ ⴷⵉⴽⵙ ⴰⵙⵉⵙⵙⵏ ⵏ ⵡⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ (ⵉⵎⴽ ⵉⴳⴰ ⵡⴰⴷⴷⴰⴷ ⵏ ⵓⵣⵎⵎⴻⵎ ⴰⵎⵔⴰⵡⵉ ⴰⵜⵔⴰⵔ), ⵉⵎⴽ ⵉⵡⴷⴰ ⴳ ⵓⵙⵎⴷⵢⴰ ⵏ ⵉⵎⴹⴰⵏ ⵉⵅⴰⵜⴰⵔⵏ.

Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero.

ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵟⴰ ⵙⴳ ⴱⵔⴰⵀⵎⴰⵙⴼⵓⵟⴰⵙⵉⴷⴰⵏⵟⴰ; ⴰⴷⵍⵉⵙ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵏⵏⴰⵏ ⴰⵎⵢⴰ ⵉⴳⴰ ⵓⵟⵟⵓⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵉⴳⴰ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵟⴰ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵙⵔⵙⵏ ⴰⵙⵉⵙⵙⵏ ⵏ ⵓⵎⵢⴰ.

In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).

ⴳ ⵓⵙⴰⵔⴰ ⴷⵖ ⵏⵏⵉⴽ ⵉⵙⵙⵎⵔⵙ ⴱⴰⵏⵉⵏⵉ ( ⴰⵙⴰⵜⵓ ⵡⵉⵙⵙ 5 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⴰⵎⵡⵓⵔⵉ ⵉⵅⵡⴰⵏ (ⴰⵎⵢⴰ), ⴳ ⴰⵛⵜⴰⴷⴰⵢⴰⵢⵉ, ⵉⴳⴰⵏ ⴰⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ ⵏ ⵉⵍⴳⴰⵎⵏ ⵍⵊⵉⴱⵔ ⵏ ⵜⵓⵜⵍⴰⵢⵜ ⵜⴰⵙⴰⵏⵙⵉⴽⵔⵉⵜ (ⵥⵕ ⴰⵡⴷ ⴱⵉⴳⴰⵍⴰ).

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals.

ⴷⴷⴰⴳ ⵢⵓⵡⴹ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 130 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵉⴽⴽⴰⵜ ⴱⴰⵟⵍⵉⵎⵓⵙ ⵉⴹⵉⵚ ⵙ ⴱⵀⵉⴱⴰⵔⵅⵓⵙ ⴷ ⵍⴱⴰⴱⵉⵍⵢⵢⵉⵏ, ⵉⵙⵡⵓⵔⵉ ⵉ ⵜⵎⴰⵜⴰⵔⵜ 0 ( ⵢⴰⵜ ⵜⵡⵔⴻⵔⵔⴰⵢⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ ⵉⵍⴰⵏ ⵢⴰⵏ ⵉⴼⵉⵍⵓ ⴰⵎⴰⴼⵍⵍⴰ ⴰⵖⵣⵣⴰⴼ ), ⴰⴳⵏⵙⵓ ⵏ ⵓⵏⴳⵔⴰⵡ ⴰⵏⵇⴹ ⴰⵡⵙⵢⴰⵏ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵢⵓⵏⴰⵏⵉⵢⵏ ⵉⴳⵎⵎⴰⵢⵏ.

Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today.

ⵉⵜⵜⵓⵎⵔⴰⵔⴰ ⵡⴰⵡⴰⵍ ⵖⴼ ⵓⵙⴰⵖⵓⵍ ⴷⵢⵓⴼⴰⵏⵜⵓⵙ ⵢⴰⴷ ⵉⵜⵜⵓⵙⵔⵙⵏ ⵙ ⵓⵙⵙⴼⵔⵓ ⵏ ⵓⵎⵓⵙⵏⴰⵡ ⵏⵜⵓⵙⵏⴰⴽⵜ ⴰⵀⵉⵏⴷⵉ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵜⴰ ⴳ ⴱⵔⴰⵀⵎⴰⵙⴼⵓⵟⴰⵙⵉⴷⴰⵏⵟⴰ ⴳ 628, ⵉⵙⵙⵎⵔⵙⵏ ⵓⵟⵟⵓⵏ ⵓⵣⴷⵉⵔⵏ ⵎⴰⵔ ⴰⴷ ⵉⵙⵏⴼⵍⴻⵍ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴽⴽⵓⵥⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ, ⵉⵙⵓⵍⵏ ⵙ ⴰⵙⵙⴰ ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙ.

At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.

ⴳ ⵜⵉⵣⵉ ⵏⵏⴰⵖ ⵏⵏⵉⴽ, ⵍⵍⴰⵏ ⵉⵚⵉⵏⵉⵢⵏ ⵙⵏⵄⴰⵜⵏ ⵙ ⵓⵟⵟⵓⵏⴻⵏ ⵓⵣⴷⵉⵔⵏ ⵙ ⵜⴱⵔⵉⴷ ⵏ ⵡⵓⵏⵓⵖ ⵏ ⵉⵣⵔⵉⵔⵉⴳ ⴰⵡⵓⵎⴰⵏ ⵙⴳ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵢⴼⴼⴰⵙ ⵓⵔ ⵉⴳⵉⵏ ⴰⵎⵢⴰ, ⵙⴳ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵉⴹⴰⵏ ⵓⵎⵏⵉⴳ ⵉⵍⵍⴰⵏ ⵏⵉⵍ ⴰⵙ.

Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory.

ⵙⴽⵔⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵉⵢⵓⵏⴰⵏⵉⵢⵏ ⴷ ⵉⵀⵉⵏⴷⵉⵢⵏⵉⴽⵍⴰⵙⵉⴽⵉⵢⵏ, ⵜⵉⵣⵔⴰⵡⵉⵏ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⵣⵓⵏⴷ ⵉⵎⵉⴽ ⴳ ⵜⵣⵔⴰⵡⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵎⴹⴰⵏ.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem.

ⵉⵣⵍⵖ ⵓⵔⵎⵎⵓⵙ ⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵔⴰⵡⵏ, ⴰⵣⵍⴰⵖ ⵉⵜⵣⵎⵎⴰⵎⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⴰⵎⵔⴰⵡ, ⵍⴰⵏ ⵙⵙⵉⵏ ⴰⴱⵓⵖⵍⵓ ⵉⵎⴰⵏⴻⵏ.

However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵉⴼⴼⵓⵍⵙ ⴼⵉⵜⴰⵖⵓⵔⵙ ⵜⵉⵍⵉⵜ ⵏ ⵉⵎⴹⴰⵏ, ⴷ ⵓⵔ ⵉⵣⴹⴰⵕ ⴰⴷ ⵉⵙⵢⴰⵀⴰ ⵏ ⵜⵉⵍⵉⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⵎⴳⵉⵏⴻⵏ.

By the 17th  century, mathematicians generally used decimal fractions with modern notation.

ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 17, ⵙⵙⵎⵔⵙⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵙ ⵓⵎⴰⵜⴰ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵔⴰⵡⵏ ⵙ ⵓⵣⵎⵎⴻⵎ ⴰⵜⵔⴰⵔ.

In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine, Georg Cantor, and Richard Dedekind was brought about.

ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1872 ⵜⵜⵓⴼⵙⴰⵔⵏⵜ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⴽⴰⵕⵍ ⵡⵢⵢⵉⵔⵙⵜⵔⴰⵙ ( ⵙⴳ ⵖⵓⵔ ⵓⵏⵍⵎⴰⴷ ⵏⵏⵙ ⵉⵢ ⴽⵓⵙⴰⴽ), ⴷ ⵉⴷⵡⴰⵕⴷ ⵀⴰⵢⵏⴰ ⴷ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ ⴷ ⵔⵉⵜⵛⴰⵔⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ.

Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties.

ⴰⵔ ⵉⵙⴽⴰⵏ ⵡⵉⵢⵉⵙⵜⵔⴰⵙ ⴷ ⴽⴰⵏⵜⵓⵔ ⴷ ⵀⵉⵏ ⵜⵉⵎⴳⵓⵏⵉⵏ ⵏⵏⵙⵏ ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵜⴰⵔⵜⵎⵉ, ⵉⵙⴱⴷⴷ ⴷⵉⴷⴽⵉⵏ ⵜⴰⵡⵏⴳⵉⵎⵜ ⵏ ⵜⵓⴱⵓⵢⵜ (ⵙⵛⵏⵉⵜ) ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵉⴱⴹⴰ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ ⴽⵓⵍ ⵅⴼ ⵙⵏⴰⵜ ⵜⵔⵓⴱⴱⴰ ⵎⵉ ⵖⵓⵔ ⵍⵍⴰⵏ ⴽⴰⵏ ⵉⴼⵕⴹⵉⵚⵏ ⵉⵎⵥⵍⴰⵢⵏ.

Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations).

ⵙⴳ ⵓⵢⴰ ⵉⵇⵏⴻⵏ ⵓⵅⵣⵣⵔ ⴳ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴰⵎⵓⵣⵣⵓⵜ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵍⵊⵉⴱⵔ ( ⵉⴼⵙⵙⴰⵢⵏ ⵎⴰⵕⵕⴰ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵎⵉ ⴳⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ).

Aristotle defined the traditional Western notion of mathematical infinity.

ⵉⵙⵏⵎⵍ ⴰⵔⵉⵙⵟⵓ ⴰⵔⵎⵎⵓⵙ ⴰⵖⵔⴱⵉ ⴰⵣⴰⵢⴽⵓ ⵖⴼ ⵜⴰⵔⵜⵎⵉ ⵜⵓⵙⵏⴰⴽⵜ.

But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis.

ⵎⴰⴽⴰ ⴰⵣⵣⵉⴳⵣ ⴰⴷ ⵎⵇⵇⵓⵕⵏ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵉⵙⴽⵔⵜ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ; ⴳ 1895 ⵉⴼⵙⵔ ⴰⴷⵍⵉⵙ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ ⵜⴰⵎⴰⵢⵏⵓⵜ ⵉ ⵜⵔⴰⴱⴱⵓⵜ, ⵉⵙⵙⵏⴽⴷ ⴰⴽⴷ ⵜⵖⴰⵡⵙⵉⵡⵉⵏ ⵏⵏⵉⴹⵏ, ⵉⵎⴹⴰⵏ ⵉⵣⵔⵔⵉⵏ ⵉⵡⵜⵜⴰ, ⴷ ⵉⵙⵙⴽⵔ ⵜⵓⵔⴷⴰ ⵜⴰⵙⵓⵍⵜ.

"A modern geometrical version of infinity is given by projective geometry, which introduces ""ideal points at infinity"", one for each spatial direction."

ⵜⵓⵏⵖⵉⵍⵜ ⵜⴰⵏⵣⴳⴰⵏⵜ ⵜⴰⵜⵔⴰⵔⵜ ⵙⴳ ⵡⴰⵔⵜⵎⵉ ⴷ ⵉⵜⵜⵓⴼⴽⴰⵏ ⵙⴳ ⵜⴰⵏⵣⴳⴰⵏⵜ ⵜⴰⵙⵜⵓⵜⵜⵉⵜ, ⵏⵏⴰⴷ ⵢⴰⴽⴽⴰⵏ ⵜⵉⵏⵇⵇⴰⴹ ⵢⴰⵜⵜⵓⵢⵏ ⵙⴳ ⵖⵓⵔ ⵡⴰⵔⵜⵎⵉ, ⵜⴰⵏⵇⵇⵉⴹⵜ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵏⵉⵍⴰ ⵜⴰⴷⵖⴰⵔⴰⵏⵜ.

The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De algebra tractatus.

ⵜⴱⴰⵢⵏⴷ ⵜⵡⵏⴳⵉⵎⵜ ⵏ ⵓⵙⵎⴷⵢⴰ ⴰⵡⵏⵖⴰⵏ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵙⴳ 1685 ⴳ ⵓⴷⵍⵉⵙ “ⵡⴰⵍⵉⵙ” “ⵉⵙⴰⴽⴰⵜⵏ ⵏ ⵍⵊⵉⴱⵔ”.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.

ⴳ 240 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵉⵙⵙⵎⵔⵙ ⵉⵔⴰⵜⵓⵙⵜⴰⵏⵙ ⴰⵔⴽⴽⵓⵜ ⵏ ⵉⵔⴰⵜⵓⵙⵜⴰⵏⵙ ⵏ ⵓⵥⵍⴰⵢ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵙ ⵣⵣⵔⴰⴱⵉⵜ.

Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes.

ⵜⵉⵢⴰⴼⵓⵜⵉⵏ ⵏⵏⵉⴹⵏ ⵏ ⵓⴱⵟⵟⵓ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷⵉⴽⵙ ⵜⵓⵣⴳⵉⵜ ⵏ ⵓⵍⵔ ⵉⵜⵜⵉⵏⵉⵏ ⵎⴰⵙⴷ ⴷⴰ ⵜⵜⵎⵄⵔⴰⵇⵏⵜ ⵜⵔⴱⵉⵄⵉⵏ ⵏ ⵜⵎⴳⴳⵉⵜⵉⵏ ⵉⵜⵜⵎⴼⴽⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷ ⵓⵙⵡⵉⵏⴳⵎ ⵏ ⴳⵓⵍⴷⴱⴰⵛ ⵏⵏⴰ ⵉⵜⵜⵉⵏⵉ ⵎⴰⵙⴷ ⴽⴰ ⵉⴳⴰⵜ ⵉⵎⵉⴹ ⴰⵅⴰⵜⴰⵔ ⵉⴳⴰⵏ ⵙⵉⵏ ⴳ ⵜⵍⵍⴰ ⵜⵓⴳⴷⵓⵜ ⵉⴳⴰ ⵜⴰⵎⵓⵏⵜ ⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ ⵙⴳ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ.

Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.)

ⵙ ⵜⵣⴰⵢⴽⵓⵜ, ⵜⵙⵙⵏⵜⵉ ⵜⵙⵏⵙⵍⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏ ⵙ ⵡⵓⵟⵟⵓⵏ 1 (0 ⵓⵔ ⴷⵊⵓⵏ ⵉⴳⵉ ⵓⵟⵟⵓⵏ ⵏ ⵍⵢⵓⵏⴰⵏ ⵉⵇⴱⵓⵔⵏ).

In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.

ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵜⵙⵉⵍⴰ ⴰⴷ 10, ⴷⴰ ⵉⵜⵜⵉⵍⵉ ⵉ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵜⵙⴳⴰ ⵏ ⵓⵢⴼⴼⴰⵙ ⵡⴰⵍⴰ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵖⴰⵔⴰⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵉⴳⴰⵏ 1, ⴷ ⵉ ⵡⵓⵟⵟⵓⵏ ⵢⴰⴹⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵎⵔⴰⵡ ⵏ ⵜⵉⴽⴽⴰⵍ ⵖⴼ ⵡⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵢⴼⴼⴰⵙ ⵏⵏⵙ.

Negative numbers are usually written with a negative sign (a minus sign).

ⴷⴰ ⵡⴰⵍⴰ ⵜⵢⴰⵔⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵓⵣⴷⵉⵔⵏ ⵙ ⵜⵎⴰⵜⴰⵔⵜ ⵜⵓⵣⴷⵉⵔⵜ ( ⵜⴰⵎⴰⵜⴰⵔⵜ ⵜⴰⵏⴰⴽⵜⴰⵎⵜ).

Here the letter Z comes .

ⴷⴰⴷⵖ ⴷⴰⴷ ⵉⵜⴷⴷⵓ ⵓⵙⴽⴽⵉⵍ ⵥ.

Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0.

ⵉⵖⵢ ⴰⴷ ⴳⵉⵏ ⵉⵎⵜⵡⴰⵍⵏ; ⵏⵏⵉⴳ ⵏ, ⵏⵖⴷ ⴷⴷⴰⵡ ⵏ, ⵏⵖ ⴰⴽⵙⵓⵍⵏ ⴷ 1, ⴷ ⵉⵖⵢ ⴰⴷ ⴳⵉⵏ ⵓⵎⵏⵉⴳⵏ ⵏⵖⴷ ⵓⵣⴷⵉⵔⵏ ⵏⵖⴷ 0.

The following paragraph will focus primarily on positive real numbers.

ⵜⵜⵡⴰⵏ ⵏ ⵜⵙⴷⴷⴰⵔⵜ ⴰⴷ ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵖⴼ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ ⵉⴳⵏ ⵓⵎⵏⵉⴳⵏ.

Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.

ⵉⵎⴽⵉ, ⵙ ⵓⵎⴷⵢⴰ, ⴰⵣⴳⵏ ⵉⴳⴰ 0.5, ⵜⵉⵙⵙ ⵙⵎⵎⵓⵙ ⵜⵡⴰⵍ ⵜⴳⴰ 0.2, ⵜⵉⵙⵙ ⵎⵔⴰⵡⵜ ⵜⵡⴰⵍ ⵜⴳⴰ 0.1, ⴷ ⵢⴰⵏ ⵅⴼ ⵙⵎⵎⵓⵙ ⵎⵔⴰⵡ ⵉⴳⴰ 0.02.

Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral.

ⵓⵔ ⵉⴷ ⵖⴰⵙ ⵉⵎⴷⵢⴰⵜⵏ ⴰⴷ ⴰⵢⴷ ⵉⵜⵢⴰⵙⵙⵏ ⴷⴰⵢ, ⵎⴰⴽⴰ ⵎⴰⵕⵕⴰ ⵓⵟⵟⵓⵏ ⵏ ⵜⵉⴷⵜ ⵓⵔ ⵡⴰⵍⴰ ⴳⵉⵏ ⵓⵎⴳⵉⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵓⵔ ⵍⵉⵏ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵜⵜⵢⴰⵍⵙⵏ, ⴰⵖⴼ ⵓⵔ ⵉⵍⵍⵉ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵔⴰⵡ ⴰⵎⴰⴽⵙⴰⵍ.

Since not even the second digit after the decimal place is preserved, the following digits are not significant.

ⵎⴰⵛⴽⵓ ⵓⵔ ⵉⵍⵍⵉ ⵓⵃⵟⵟⵓ ⵏ ⵡⵉⵙⵙ ⵙⵉⵏ ⴹⴰⵕⵜ ⵜⵉⵙⴽⵔⵜ ⵜⴰⵎⵔⴰⵡⵜ, ⴰⵖⴼ ⵓⵔ ⵢⴰⴷ ⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⵡⵓⵟⵟⵓⵏ ⴷ ⵉⴽⴽⴰⵏ ⴷⴰⵜ.

For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1.

ⵙ ⵓⵎⴷⵢⴰ, 0.999..., 1.0, 1.00, 1.000, ..., ⴽⵓⵍⵍⵓⵜⵏ ⵙⵎⴷⵢⴰⵏ ⵙ ⵡⵓⵟⵟⵓⵏ ⴰⵖⴰⵔⴰⵏ 1.

Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place.

ⵜⵉⵢⵉⵔⴰ, ⵉⴳ ⵍⵍⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵎⴰⵕⵕⴰ ⴳ ⵡⵓⵟⵟⵓⵏ 0, ⴷⴰ ⵉⵜⴳⴳⴰ ⵡⵓⵟⵟⵓⵏ 0, ⴷ ⵎⴽ ⴳⴰⵏ ⵡⵓⵟⵟⵓⵏ ⴽⵓⵍ ⴳ ⵡⵓⵟⵟⵓⵏ ⵏ ⵜⴳⴼⴼⵓⵔⵜ ⵓⵔ ⵉⵜⴼⵓⴽⴽⵓⵏ ⴳ 9, ⵜⵖⵉⴷ ⴰⴷ ⵜⵙⵓⴳⵣⵣⴷ ⵜⵥⴰ ⴳ ⵓⵢⴼⴼⴰⵙ ⴳ ⵓⴷⵖⴰⵔ ⴰⵎⵔⴰⵡ, ⵜⵔⵏⵓⴷ ⵢⴰⵏ ⵉ ⵜⴳⴼⴼⵓⵔⵜ ⴳ ⵍⵍⴰⵏ 9 ⵏ ⵜⵉⵙⵏⴰⵜⵉⵏ ⴳ ⵓⵥⵍⵎⴰⴹ ⵏ ⵡⴰⵏⵙⴰ ⴰⵎⵔⴰⵡ.

Thus the real numbers are a subset of the complex numbers.

ⴷ ⴳⵉⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ; ⵜⴰⵔⴱⵉⵄⵜ ⵜⴰⵢⵢⴰⵡⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ,

The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a root in the complex numbers.

ⵜⵙⵍⴽⴰⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵏ ⵍⵊⵉⴱⵔ ⵖⴼ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵉⵜⵜⴳⴳⴰⵏ ⵉⴳⵔ ⵏ ⵍⵊⵉⴱⵔ ⵉⴳⵏ ⴰⵎⴰⵖⵓⵏ, ⴰⵖ ⵉⵜⵜⵉⵏⵉⵏ ⵉⵙⴷ ⴰⵎⴳⴳⵓⴷⵢ ⵏ ⵉⵡⵜⵜⴰ ⴰⴽⴷ ⵉⵔⵡⵉⵏ ⵖⵓⵔⵙ ⴰⵥⵓⵕ ⴳ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ.

The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered.

ⵜⵜⵓⴳⴰ ⵜⵣⵔⴰⵡⵜ ⵖⴼ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⴳ ⵓⴼⵓⵖⴰⵍ ⴰⵎⵉⵔⵉⵡ ⵓⴳⴳⴰⵔ ⵏ 2000 ⵏ ⵓⵙⴳⴳⵯⴰⵙ, ⵢⵓⵡⵉⵏ ⵖⵔ ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⵇⵙⵉⵜⵏ, ⵜⵜⵓⴼⴽⴰ ⵜⵎⵔⴰⵔⵓⵜ ⵉ ⵉⵜⵙⵏ ⴷⴰⵢ.

Real numbers that are not rational numbers are called irrational numbers.

ⴷⴰ ⵜⵙⵎⵎⴰⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵏⵏⴰ ⵓⵔ ⵉⴳⵉⵏ ⵓⵟⵟⵓⵏⴻⵏ ⵓⵎⴳⵉⵏⴻⵏ; ⵉⵎⴹⴰⵏ ⵓⵔ ⵉⴳⵉⵏ ⵓⵎⴳⵉⵏⴻⵏ.

The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers.

ⵓⵟⵟⵓⵏ ⵉⵜⵜⵓⵙⵉⵟⵏ; ⵡⵔⵏ ⵖⴼ ⵜⵉⴳⴳⵉⵜⵉⵏ ⵏ ⵓⵙⵙⵉⵟⵏ ⵉⵜⵜⵓⵡⴰⵍⴼⵏ, ⴳ ⵉⵍⵍⴰ ⵓⵙⵙⵉⵟⵏ ⵏ ⵓⵥⵓⵕ ⵏ ⴱⵓ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵜⵙⴽⴰⵔ ⵉⴳⵔ ⴰⵎⴰⵇⵇⴰⵏ ⵏ ⵜⵉⴷⵜ ⴳ ⵍⵍⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵏ ⵍⵊⵉⴱⵔ ⵏ ⵜⵉⴷⵜ.

One reason is that there is no algorithm for testing the equality of two computable numbers.

ⵢⴰⵏ ⴳ ⵉⵎⵏⵜⵉⵍⵏ ⵉⴳⴰⵜ ⵓⵔ ⵜⵍⵍⵉ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵢⵉⵔⵎ ⵏ ⵜⵏⴳⵉⴷⴷⵉⵜ ⵉⵏⴳⵔ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⵉⴳⵏ ⵡⵉⵏ ⵓⵙⵙⵉⵟⵏ.

The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.

ⴷⴰ ⵉⵙⴽⵓⵜⵜⵓ ⵓⵏⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ ⴷ ⵉⵜⵜⵓⴼⴽⴰⵏ ⵙ ⵓⵙⵉⵍⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵉ ⵡⵓⵟⵟⵓⵏ, ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵍⴳⴰⵎⵜ ⵉⵖⵉⵏ ⴰⴷ ⵢⵉⵍⵉ, ⵎⴰⴽⴰ ⵜⴰⵍⴳⴰⵎⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷⴰ ⵜⴰⴽⴽⴰ ⵉⵎⵥⵍⴰⵢ ⵉⵖⵓⴷⴰⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.

The former gives the ordering of the set, while the latter gives its size.

ⴷⴰ ⵢⴰⴽⴽⴰ ⵓⵎⵣⵡⴰⵔⵓ ⴰⵙⵙⵓⴷⵙ ⵏ ⵜⵔⴱⵉⵄⵜ, ⴰⵔ ⵢⴰⴽⴽⴰ ⵓⵎⴳⴳⴰⵔⵓ ⴰⴽⵙⴰⵢ ⵏⵏⵙ.

This standard basis makes the complex numbers a Cartesian plane, called the complex plane.

ⴰⵙⵉⵍⴰ ⴰⴷ ⴰⵎⵙⵖⴰⵍ ⴷⴰ ⵉⵜⴳⴳⴰ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⴳ ⵓⵙⵡⵉⵔ ⴰⴷⵉⴽⴰⵔⵜⵉⵢ ⴷⴰ ⵉⵜⵙⵎⵎⴰ ⴰⵙⵡⵓⵔ ⵓⴷⴷⵉⵙ.

The complex numbers of absolute value one form the unit circle.

ⴷⴰ ⵜⴳⴳⴰⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵏ ⵡⴰⵜⵉⴳ ⴰⵎⴳⴳⴰⵔⵓ, ⵢⴰⵜ ⵜⵣⴳⵓⵏⵜ ⵉⴳⵏ ⵢⵓⵡⵜ.

In domain coloring the output dimensions are represented by color and brightness, respectively.

ⴳ ⵢⵉⴳⵔ ⵏ ⵓⵙⵓⵖⵏ, ⴷⴰ ⵜⵜⵓⵙⵎⴷⵢⴰⵏ ⵡⵓⴳⴳⵓⴳⵏ ⵏ ⵓⵙⵙⵓⴼⵖ ⵙ ⵓⴽⵯⵍⵉ ⴷ ⵓⵣⵣⵏⵥⵕ ⴰⵎⵣⴷⴰⵢ.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

ⵢⵓⵡ ⵓⵙⵡⵓⵔⵉ ⵖⴼ ⵜⵎⵓⴽⵔⵉⵙⵉⵏ ⵏ ⵜⵎⴳⴳⵓⴷⵉⵏ ⵏ ⵉⵡⵜⵜⴰ ⵜⵉⵎⴰⵜⵜⵓⵜⵉⵏ ⴳ ⵜⵢⵉⵔⴰ ⵖⵔ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵙⵉⵍⴰⵏⵜ ⵏ ⵍⵊⵉⴱⵔ, ⵏⵏⴰ ⵉⵙⵙⴼⵔⵓⵏ ⴰⴽⴷ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ, ⵉⵍⵍⴰ ⵓⴼⵙⵙⴰⵢ ⵏ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵉⵍⴰ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ; ⵙⴳ ⵜⵙⴽⵯⴼⵍⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵏⵖⴷ ⵏⵏⵉⴳ ⴰⵙ.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.

ⴱⴰⵢⵏⴷ ⵜⵙⵎⴽⵜⵉⵜⵉⵏ ⵏ ⵡⵉⵙⵍ ⴳ ⵜⵉⵡⵉⵏ ⵜⵉⴽⴰⴷⵉⵎⵉⵜⵉⵏ ⵣⵓⵏⴷ “ⴽⵓⴱⵏⵀⴰⴳⵏ”, ⵎⴰⴽⴰ ⵜⵣⵔⵉ ⵓⵔ ⵜⵜ ⵢⴰⵏⵏⴰⵢ ⴰⵡⴷ ⵢⴰⵏ ⴽⵉⴳⴰⵏ.

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.

ⵙⴳ ⵉⵎⴰⵔⵔⴰⵜⵏ ⵉⴽⵍⴰⵙⵉⴽⵏ ⴳ ⵜⵉⵣⵉ ⵢⴰⴹⵏ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵏ ⵔⵉⵜⵛⴰⵔⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ, ⵏⵖⴷ ⵓⵜⵓ ⵀⵉⵍⴷⵔ,ⴷ ⴼⵉⵍⵉⴽⵙ ⴽⵍⵉⵏ, ⴷ ⵀⵉⵏⵔⵉ ⴱⵡⴰⵏⴽⵉⵔ, ⴷ ⵀⵉⵔⵎⴰⵏ ⵛⵡⴰⵔⵜⵣ, ⴷ ⴽⴰⵔⵍ ⵡⵉⵔⵙⵜⵔⴰⵙ ⴷ ⵡⵉⵢⵢⴰⴹ.

The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).

ⵓⵔ ⵉⵜⵜⵓⵙⵢⴰⵀⴰ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵎⴹⴰⵏ ⵉⵡⵏⴳⵉⵎⵏ ⴳ ⵓⵖⵓⴼⴰⵍ ⴰⵎⵉⵔⵉⵡ, ⴰⵔ ⵜⴰⵡⵓⵔⵉ ⵏ ⵍⵢⵓⵏⴰⵔⴷ ⵓⵍⵔ (1707-1783), ⴷ ⴽⴰⵕⵍ ⴼⵔⵉⴷⵔⵉⵜⵛ ⴳⴰⵡⵙ (1777-1855).

The integers form the smallest group and the smallest ring containing the natural numbers.

ⴷⴰ ⵜⴳⴳⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ; ⵜⴰⵔⴱⵉⵄⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ, ⴷ ⵜⵅⵔⵙⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ ⴳ ⵍⵍⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏⴻⵏ.

It is the prototype of all objects of such algebraic structure.

ⵉⴳⴰ ⴰⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ ⵏ ⵎⴰⵕⵕⴰ ⵉⵎⵖⵏⴰⵡⵏ, ⵣⵓⵏⴷ ⵜⵓⵙⴽⵉⵡⵜ ⵜⴰⵊⵉⴱⵔⵉⵜ.

Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).

ⴷⴰ ⵜⵜⵓⴱⴷⴰⵔⵏ ⵡⴰⵏⴰⵡⵏ ⵏ ⵉⵏⵎⵎⴰⵍⵏ ⵏ ⵓⵙⵏⵎⵉⵍⵉ ⵏ ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⵉⵍⴰⵏ ⵜⴰⵖⵣⵉ ⵉⵡⵔⵏ (ⵏⵖⴷ ⵜⵉⵔⴱⵉⵄⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ), ⵉⴳⴰⵏ ⵙ ⵜⵎⴰⵜⴰⵔⵜ “ int “ ⵏⵖⴷ “Integer” ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⵓⵜⵍⴰⵢⵉⵏ ⵏ ⵓⵙⵖⵉⵡⵙ ( ⵣⵓⵏⴷ Algol68, ⴷ C, ⴷ Java,ⴷ Delphi, ⴷ ⵜⵉⵢⵢⴰⴹ).

These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.

ⵉⵎⵥⵍⴰⵢ ⴰⴷ ⵉⵖⵢ ⴰⴷ ⵜⵏⵜ ⵏⵙⵡⵔ ⵉ ⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⴷ ⵉⵏⴳⵔⴰⵡⵏ ⵉⵎⵉⴹⴰⵏ ⵉⵎⵙⵓⵔⵙⵏ, ⴷ ⵓⵔⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙ ⴰⵎ ⵉⵙⵉⵙⵙⵏ.

Since the triangle is isosceles, a = b).

ⵎⴰⵢⴷ ⵉⴳⴰ ⵡⴰⵎⴽⵕⴰⴹ ⴰⵎⵙⴰⵙⴽⵍ ⵏ ⵉⴳⴰⵍⴰⵍⵏ, a = b).

Since c is even, dividing c by 2 yields an integer.

ⵎⴰⵢⴷ ⵉⴳⴰ c ⴰⵎⵙⵉⵏ, ⴷⴰ ⴰⵖⴷ ⵜⴰⴽⴽⴰ ⵜⵓⴱⴹⵓⵜ ⵏ c ⵅⴼ 2; ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ.

Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.

ⴰⵙⵏⴼⵍ ⵏ 4y2 ⵖⵔ c2, ⴳ ⵜⴳⴰⴷⴰⵣⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ (c2 = 2b2), ⴰⵖ ⵢⴰⴽⴽⴰⵏ 4y2= 2b2.

Since b2 is even, b must be even.

ⵎⴰⵢⴷ ⵉⴳⴰ b2 ⴰⵎⵙⵉⵏ, ⵉⵇⵏⴻⵏ ⴰⴷ ⵉⴳ b ⴰⵎⵙⵉⵏ.

However this contradicts the assumption that they have no common factors.

ⵎⴰⴽⴰ ⴰⵢⴰ ⵉⵜⵎⴳⵍⴰ ⴷ ⵓⵎⵔⴷⵓ ⵉⵜⵜⵉⵏⵉⵏ ⵓⵔ ⵍⵍⵉⵏ ⵉⵎⴳⴳⵉⵜⵏ ⵉⵎⵛⵛⵓⵔⵏ ⴳⵔⴰⵙⵏ.

Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ , ⵉⵍⵍⴰ ⵢⵉⵍⵖ ⵏ ⵀⵉⴱⴰⵙⵓⵙ ⵖⴼ ⵜⵣⵎⵎⴰⵔ ⵏⵏⵙ, ⴰⵢⴷ ⵉⵏⵏⴰ ⵢⴰⵏ ⵡⵓⵎⵉⵢ, ⵢⵓⴽⵣ, ⵉⴽⵉⵣ ⵏⵏⵙ ⴰⵢⵏⵏⴰ ⵉⵍⵍⴰ ⴳ ⵢⵉⵍ, ⴳⵔⵏⵜ ⵉⵎⴷⴷⵓⴽⴰⵍ ⵏⵏⵙ ⵉⴼⵉⵜⴰⵖⵓⵔⵙⵏ ⴳ ⵜⵉⵣⵉ ⵢⴰⴹⵏ,..., ⴰⵛⴽⵓ ⵙⵏⴼⵍⵏ ⵢⴰ ⵓⴼⵕⴹⵉⵚ ⴳ ⵉⵖⵣⵡⵔ ⵓⵔ ⵉⵔⵉⵏ... ⵜⴰⵖⴰⵍⵜ ⵉⵜⵜⵉⵏⵉⵏ ⵉⵖⵢ ⵓⵣⴳⵣⵍ ⵏ ⵜⵓⵎⴰⵏⵉⵏ ⴳ ⵉⵖⵣⵡⵔ ⵖⵔ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⴷ ⵓⵙⵖⴰⵍ ⵏⵏⵙⵏ.

For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on.

ⵙ ⵓⵎⴷⵢⴰ, ⴳ ⵜⵡⵏⴳⵉⵎⵜ ⵏⵏⴽ ⵜⴰⴳⵣⵣⵓⵎⵜ ⵉⵏⵎⵏ: ⵉⵖ ⴰⴷ ⵜⵜⵓⴱⴹⵓ ⵖⵔ ⵙⵏⴰⵜ, ⴷ ⵓⵣⴳⵏ ⵏ ⵓⵣⴳⵏ ⵖⵔ ⵙⵉⵏ, ⴷ ⵓⵣⴳⵏ ⵏ ⵓⵣⴳⵏ ⵖⴼ ⴰⵣⵏⴳ, ⵉⵎⴽⵉ.

This is just what Zeno sought to prove.

ⴰⵢⴰ ⴰⵢⴷ ⵉⵜⵏⴰⵖ ⵣⵉⵏⵓⵏ ⴰⴷ ⵜⵜ ⵉⵙⵡⵔ.

In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.

ⴳ ⵜⵡⵏⴳⵉⵎⵉⵏ ⵏ ⵍⵉⵖⵔⵉⵇ, ⵓⴽⵓⵙ ⵏ ⵓⵎⴷⴷⴰⴷ ⵏ ⴽⴰⵏ ⵜⴰⵏⵏⴰⵢⵜ ⵓⵔ ⴷⴰ ⵉⵙⵡⴰⵔ ⴰⵎⴷⴷⴰⴷ ⵏ ⵜⴰⵏⵏⴰⵢⵜ ⵏⵏⵉⴹⵏ, ⵅⴼ ⵓⵢⴰ ⴰⵙⵏ ⵉⵇⵇⵏ ⵓⵣⵔⵔⵓ ⵓⴳⴳⴰⵔ.

A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously.

ⴰⴽⵙⴰⵢ”...ⵓⵔ ⵉⴳⵉ ⵉⵎⵉⴹ, ⵎⴰⴽⴰ ⵉⵙⵎⴷⵢⴰ ⵙ ⵉⵏⵎⴰⵍⵍⴰⵜⵏ ⵣⵓⵏⴷ: ⵉⵎⵣⵔⴰⵢⵏ ⵏ ⵓⵣⵔⵉⵔⵉⴳ, ⴷ ⵜⵖⵎⵔⵉⵏ, ⴷ ⵜⵉⵊⵉⵎⵎⴰ, ⴷ ⵉⴽⵙⴰⵢⵏ, ⴷ ⵜⵉⵣⵉ ⵉⵖⵉⵏ ⴰⴷ ⵜⵎⵣⵉⵔⵉⵢ, ⵉⵎⴽ ⵏⵜⵜⵉⵏⵉ ⵙ ⵜⴰⵍⵖⴰ ⵉⵣⴷⵉⵏ.

Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios.

ⴰⵛⴽⵓ ⵓⵔ ⵉⵜⵢⴰⴽⵣ ⵡⴰⵜⵉⴳ ⵏ ⵡⴰⵏⵛ ⵉⵍⵍⴰⵏ ⴳ ⵓⴽⵙⴰⵢ, ⵉⵖⵢ ⵉⴷⵓⴽⵙⵓⵙ ⴰⴷ ⵉⵙⵙⵉⵟⵏ ⴰⵙⵖⴰⵍⵏ ⵉⴳⴰⵏ ⵡⵉⵏ ⵓⵙⵇⵇⵓⵍ, ⴷ ⵜⵉⵏⵏⴰ ⵓⵔ ⵉⴳⵉⵏ ⵜⵉⵏ ⵓⵙⵇⵇⵓⵍ ⵙⴳ ⵓⵙⵜⴰⵢ ⵏ ⵓⵙⵖⵍ ⵙⴳ ⵓⴽⵙⴰⵢ, ⴷ ⵓⵙⵖⵍ ⵣⵓⵏⴷ ⴰⵙⵙⵉⴽⵙⵍ ⵏ ⵙⵉⵏ ⵉⵙⵖⴰⵍ.

This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.

ⴷⴰ ⵉⵜⵜⵓⴳⴰ ⵉ ⵡⵉⵏⵏⴰ ⵓⵔ ⵉⵥⴹⴰⵕⵏ ⵉ ⵓⵙⵇⵇⵓⵍ ⴳ ⵉⴼⵕⴹⵉⵙⵏ ⵏ ⵉⵇⵍⵉⴷⵙ, ⴰⴷⵍⵉⵙ ⵎⵔⴰⵡ, ⴰⵙⵓⵎⵔ 9.

In fact, in many cases algebraic conceptions were reformulated into geometric terms.

ⴳ ⵜⵉⵏⴰⵡⵜ; ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⴷⴷⴰⴷⵏ ⵉⵜⵜⵓⵢⴰⵍⵍⵙ ⵜⵉⵀⴽⵛⵜ ⵉ ⵉⵔⵎⵎⵓⵙⵏ ⵏ ⵍⵊⵉⴱⵔ ⵖⵔ ⵉⵔⵎⴰⵏ ⵉⵏⵣⴳⴰⵏⴻⵏ.

The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory.

ⵉⵇⵏⴻⵏ ⵡⵓⵙⵓⵏ ⵏ ⵉⵜⵙⵏ ⵉⵔⵎⵎⵓⵙⵏ ⵉⴷⵙⵍⴰⵏ ⴰⴳⵏⵙⵓ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⴷⵖⵉ, ⵉⵎⵣⴰⵔⴰⵢ ⵏ ⵜⵉⵍⵉⵜ ⵏ ⵜⵉⴳⴳⵉ ⴰⵣⵔⵔⵓ ⴰⴽⴽⵯ ⵉⴳⵏ ⴰⵖⵣⵓⵔⴰⵏ ⵉ ⵜⵉⴳⴳⵉⵡⵉⵏ ⴷ ⵉⵙⵡⵉⵏⴳⵎⵏ ⵉⵍⵍⴰⵏ ⴹⴰⵕⵜ ⵜⵎⴰⴳⵓⵏⵜ ⴰⴷ.

"However, historian Carl Benjamin Boyer writes that ""such claims are not well substantiated and unlikely to be true""."

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵢⴰⵔⴰ ⵓⵏⵎⵣⵔⵓⵢ ⴽⴰⵕⵍ ⴱⵏ ⵊⴰⵎⵉⵏ ⴱⵡⵉⵔ ⴰⵏ; “ⵣⵓⵏⴷ ⵜⴰⴼⴰⴽⵓⵍⵜ ⴰⴷ ⵓⵔ ⵉⵍⵉⵏ ⵉⵙⵡⵔⵏ ⵉⵖⵓⴷⴰⵏ, ⵓⵔ ⵜⵏⵏⵉ ⴰⴷ ⵜⴳ ⵜⴰⵎⴷⴷⴰⴷⵜ”.

Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.

ⴽⴰⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵣⵓⵏⴷ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵜⴰ (ⴰⵙⴳⴳⵯⴰⵙ ⵏ 628 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⴷ ⴱⴰⵙⴽⴰⵔⴰ (ⴳ 629 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⵉⵎⵢⵉⵡⴰⵙⵏ ⴳ ⵢⵉⴳⵔ ⴰⴷ, ⵎⴽ ⵙⴽⵔⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵢⴰⴹⵏ ⵉⴹⴼⴰⵕⵏ ⵎⴰⵢⴰⵏ.

The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind.

ⴳ ⵓⵙⴳⴳⵯⴰⵙ 1872 ⵜⵢⴰⴼⵙⴰⵔⵏⵜ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⴽⴰⵕⵍ ⵡⵉⵢⵔⵙⵜⵔⴰⵙ ( ⵙⴳ ⵖⵓⵔ ⵓⵏⵍⵎⴰⴷ ⵏⵏⵙ ⵉⵔⵏⵙⵜ ⴽⵓⵙⴰⴽ) ⴷ ⵉⴷⵡⴰⵕⴷ ⵀⴰⵢⵏ ( ⵜⴰⵙⵖⵏⵜ ⵏ ⴽⵕⵉⵍ, 74), ⴷ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ (ⴰⵏⴰⵍⵉⵏ, 5), ⴷ ⵔⵉⵜⵛⴰⵕⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ.

Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain characteristic properties.

ⵉⵙⴽⴰ ⵡⵉⵢⵔⵙⵜⵔⴰⵙ ⴷ ⴽⴰⵏⵜⵓⵔ ⴷ ⵀⵉⵏ; ⵜⵉⵎⴰⴳⵓⵏⵉⵏ ⵏⵏⵙⵏ ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵜⴰⵔⵜⵎⵉ, ⵉⵙⴱⴷⴷ ⴷⵉⴷⴽⵉⵏ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ ⵖⴼ ⵜⵡⵏⴳⵉⵎⵜ ⵏ ⵡⵓⴱⵓⵢ (ⵙⵉⵛⵏⵉⵜ), ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵎⴰⵕⵕⴰ ⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⴰⵔⵜⵏ ⵢⴰⵟⵟⵓ ⵖⴼ ⵙⵏⴰⵜ ⵜⵔⵓⴱⴱⴰ ⵉⵍⴰⵏ ⵉⵜⵙⵏ ⵉⵎⵥⵍⴰⵢⵏ ⵉⵙⵜⵉⵏ.

Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

ⵉⵔⵏⴰ ⴰⵡⴷ ⴷⵉⵍⵉⵛⵍⵉⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ, ⵉⵎⴽ ⴳⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⵉⵏⵏⴰ ⵢⵓⵡⵙⵏ ⵏ ⵜⵙⵏⵙⵉⵜⵉⵏ ⵏ ⵓⵙⵏⵜⵍ.

This asserts that every integer has a unique factorization into primes.

ⴰⵢⴰ ⵉⵙⵍⴽⴰⵏ ⵏ ⵡⵉⵙ ⴷ ⴽⵓ ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⵖⴰⵔⵙ ⵉⵎⴳⴳⵉ ⴰⵎⵥⵍⴰⵢ ⴳ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ.

To show this, suppose we divide integers n by m (where m is nonzero).

ⵎⴰⵔ ⴰⴷ ⵏⵙⴱⴰⵢⵏ ⴰⵢⴰⴷ ⵢⵓⵔⴷⴰ ⵉⵙ ⴷⴰ ⵏⴰⵟⵟⵓ ⵉⵎⴹⴰⵏ ⴰⵎⴷⴷⴰⴷⵏ n ⵅⴼ ⵎ ( ⴰⵛⴽⵓ m ⵓⵔ ⵜⴳⵉ ⵜⴰⵎⵢⴰⵜ).

If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once.

ⵉⴳ ⵓⵔ ⴰⴽⴽⵯ ⵉⵜⵜⵓⴳⴰ 0, ⵜⵖⵢ ⴰⴷ ⵜⵙⵡⵓⵔⵉ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵖⴼ ⴽⵉⴳⴰⵏ ⵏ ⵜⵙⵓⵔⵉⴼⵉⵏ m - 1, ⴱⵍⴰ ⵏⵙⵙⵎⵔⵙ ⴰⵡⴷⵢⴰⵏ ⵓⵎⴰⴳⵓⵔ ⵡⴰⵀⵍⵉ ⵏ ⵜⵉⴽⴽⴰⵍ.

"In mathematics, the natural numbers are those used for counting (as in ""there are six coins on the table"") and ordering (as in ""this is the third largest city in the country"")."

“ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵓⵟⵟⵓⵏⴻⵏ ⵉⵖⴰⵔⴰⵏⴻⵏ ⴳⴰⵏⵜⵏ ⵜⵉⵏⵏⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵓⵙⵙⵉⵟⵏ ( ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ : ⵍⵍⴰⵏⵜ ⵙⴹⵉⵚ ⵏ ⵉⴷⵔⵉⵎⵏ ⴰⴼⵍⵍⴰ ⵏ ⵜⴷⴰⴱⵓⵜ), ⴷ ⵓⵙⵙⵓⴷⵙ ( ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ : ⵡⴰⴷ ⴰⵖⵔⵎ ⵡⵉⵙⵙ ⴽⵕⴰⴹ ⴳ ⵜⵎⵓⵔⵜ).”

These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.

ⴷⴰ ⵜⴳⴳⴰⵏⵜ ⵜⴳⴼⴼⵓⵔⵉⵏ ⵏ ⵉⵣⴷⴰⵡⵏ ⵏ ⵡⵓⵟⵟⵓ ⵉⵖⴰⵔⴰⵏ ⴳ ⵍⵍⴰⵏ ( ⵉⵥⵍⵉⵏ), ⴳ ⵉⵎⴰⴳⵔⴰⵡⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵢⴰⴹⵏ.

The first major advance in abstraction was the use of numerals to represent numbers.

ⴰⵣⵡⴰⵔ ⴰⵎⵣⵡⴰⵔⵓ ⵏ ⵜⴰⵏⵣⵖⵜ, ⵉⴳⴰⵜ ⵓⵙⵙⵎⵔⵙ ⴰⵎⴰⵟⵟⵓⵏ ⵉ ⵓⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ.